"That's our fourth Capital One Cup game away from home so whoever we get and when the night comes we'll apply ourselves as we did tonight," Sky Sports News quotes Mowbray as saying.
"I am delighted for the team and the supporters. It was a decent performance.
"Cup competition is about getting through to the next round and we managed to do that. I think there will be a lot of happy people on Teesside."
Mowbray went on to back his opposite number Martin O'Neill to bounce back from the defeat and turn Sunderland's fortunes around.
"I have total respect for Martin and his team," he added. "I'm sure he's scratching his head as to why they've not scored as many goals as he'd want.
"They've got a lot of attacking prowess and a lot of players who can score goals. I'm sure Martin will find the answer to the conundrum he's got at the moment and find more goals."
Middlesbrough are one of three non-Premier League sides to progress to the fifth round of the competition along with Leeds United and Bradford City.